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Lochs' Constant


For a real number x in (0,1), let m be the number of terms in the convergent to a regular continued fraction that are required to represent n decimal places of x. Then Lochs' theorem states that for almost all x,

L=lim_(n->infty)m/n
(1)
=(6ln2ln10)/(pi^2)
(2)
=0.97027014...
(3)

(OEIS A086819; Lochs 1964). This number is sometimes known as Lochs' constant.

The reciprocal of this constant is

L^(-1)=(pi^2)/(6ln2ln10)
(4)
=1.03064083410...
(5)

(OEIS A062542; Finch 2003, p. 60).

Lochs' constant is related to the Lévy constant e^beta by

L=1/(2log_10(e^beta))
(6)
=(ln10)/(2beta).
(7)

In the index and table of constants Finch (2003, pp. 546 and 596) refers to the quantity

 3/4-(3ln2)/(pi^2)(3ln2-(24zeta^'(2))/(pi^2)+4gamma-2)-(6ln2)/(pi^2)(6/(pi^2)zeta^'(2)-1/2) 
 =0.2173242870...
(8)

related to Porter's constant as "Lochs' constant," though this terminology appears to be nonstandard.


See also

Lévy Constant, Lochs' Theorem, Porter's Constant, Regular Continued Fraction

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References

ReferencesBosma, W.; Dajani, K.; and Kraaikamp, C. "Entropy and Counting Correct Digits." Univ. Nijmegen Math. Report 9925, 1999.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Kintchine, A. "Zur metrischen Kettenbruchtheorie." Compos. Math. 3, 276-285, 1936.Kraaikamp, C. "A New Class of Continued Fraction Expansions." Acta Arith. 57, 1-39, 1991.Lévy, P. "Sur le developpement en fraction continue d'un nombre choisi au hasard." Compos. Math. 3, 286-303, 1936.Lochs, G. "Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch." Abh. Hamburg Univ. Math. Sem. 27, 142-144, 1964.Perron, O. Die Lehre von den Kettenbrüchen, 3. verb. und erweiterte Aufl. Stuttgart, Germany: Teubner, 1954-57.Sloane, N. J. A. Sequences A062542 and A086819 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Lochs' Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LochsConstant.html

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